How to make part of rigged mesh stationary?
As you can read in edit mode, your base vertices do have some weight relatively to many bones, so of course they are moving.

If you want them to stay still, select all of them and remove weight from all vertex groups.

Another more usual way is to create a "tentacle base" bone and weight them to it.

Remember that even a very little weight, if it's not "counterbalanced" by a bigger weight, will have a great influence to the vertex, so you can't trust the blue color in weight paint mode only.